Improved Ramsey-type theorems for Fibonacci numbers and other sequences

TL;DR

This paper improves bounds on Fibonacci-based variants of van der Waerden numbers, exploring monochromatic progressions with Fibonacci differences and providing computational data for related sequences.

Contribution

It introduces improved bounds for Fibonacci difference variants of van der Waerden numbers and offers computational data for other difference sets.

Findings

Enhanced bounds for Fibonacci difference van der Waerden numbers.

Existence results for Fibonacci-based monochromatic progressions.

Computational data for difference sets beyond Fibonacci numbers.

Abstract

Van der Waerden's theorem states that for any positive integers $k$ and $r$, there exists a smallest value $n = w(k,r)$, called the van der Waerden number, such that every $r$-coloring of $\{1,\dots,n\}$ contains a monochromatic $k$-term arithmetic progression. We consider two variants of van der Waerden numbers: the numbers $n = n(AP_D,k;r)$, the smallest value where every $r$-coloring of $\{1,\dots,n\}$ contains a monochromatic $k$-term arithmetic progression with common difference in $D$, and the numbers $n = \Delta(D,k;r)$, the smallest value $n$ where every $r$-coloring of $\{1,\dots,n\}$ contains a sequence $x_1 < \dots < x_k$ where the differences between consecutive terms are members of $D$. We study the case when $D$ is set of Fibonacci numbers $F$ and give improved bounds for the largest $r$ where $n(AP_F,k;r)$ and $\Delta(F,k;r)$ exist for all $k$. Moreover, we give some computational data on $\Delta(D,k;r)$ for other sets $D$.